This amplification may also give rise to the formation of PBHs when the perturbations re-enter the horizon, soon after the end of inflation. Therefore, we only have light, evaporating PBHs with masses \(M_{PBH} \lesssim 10^6\)  g. They may give rise to Planck relics that may overclose the universe, and to avoid this their mass fraction is limited to values below \(10^{-16}\) . Although we have not checked case by case the PBH variance function \(\sigma (M)\)  and then the mass fraction, by looking at some examples it seems a reasonable assumption only to exclude models with \(P_{{\cal R}\, end} \lesssim 0.1\) . Scanning over parameter values compatible with primordial spectrum observable values and PBHs constraint, we obtain Fig. (REF ). The maximal amplitude is obtained at frequencies \(f \simeq 0.1\)  MHz, outside the range of detection of GW detectors, but close for example to the typical stochastic spectrum obtained during preheating after inflation [1]}, [2]}, [3]}, [4]}, [5]}. We notice that we do not expect any additional contribution due to preheating, i.e., we do not expect any parametric resonance or additional particle production to take place at the end of inflation. For the parameter space that amplify the spectrum in our case, inflation ends in a smooth transition from inflation to a radiation dominated universe, with a damped, subdominant oscillating inflaton. In any case, our mechanism adds to the possible cosmological sources of a background of gravitational sources beyond MHz frequencies, frequencies for which no astrophysical source has been identified, and which may be searched for in future detectors [6]}.
